((2207−(1/(2207−(1/(2207−(1/(2207−(1/(2207−...))))))))))^(1/(8 ))


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 128264 by bobhans last updated on 06/Jan/21

$\sqrt[8]{2207 - \frac{1}{2207 - \frac{1}{2207 - \frac{1}{2207 - \frac{1}{2207 - . . .}}}}}$
Commented by MJS_new last updated on 06/Jan/21

$\frac{3 + \sqrt{5}}{2}$
Commented by bobhans last updated on 06/Jan/21

$w a w . . . . h o w ?$
Commented by liberty last updated on 06/Jan/21

$\sqrt[8]{2207 - \frac{1}{2207 - \frac{1}{2207 - \frac{1}{2207 - . . .}}}} = \frac{a + b \sqrt{c}}{d}; a, b, c, d \in Z$ $let x = \sqrt[8]{2207 - \frac{1}{x^{8}}} \Rightarrow x^{8} = 2207 - \frac{1}{x^{8}}$ $\Rightarrow x^{16} - 2207 x^{8} + 1 = (x^{8} + {p x}^{4} + 1) (x^{8} - {p x}^{4} + 1)$ $w e f i n d p^{2} = 2209 \to p = 47$ $\Rightarrow x^{16} - 2207 x^{8} + 1 = (x^{8} + 47 x^{4} + 1) (x^{8} - 47 x^{4} + 1)$ $n o w x^{8} + 47 x^{2} + 1 = (x^{4} + {q x}^{2} + 1) (x^{4} - {q x}^{2} + 1)$ $w e g e t q = 7$ $x^{8} + 47 x^{2} + 1 = (x^{4} + 7 x^{2} + 1)$ $t h e n x^{16} - 2207 x^{8} + 1 = (x^{8} + 47 x^{4} + 1) (x^{4} + 7 x + 1) (x^{2} + 3 x + 1) (x^{2} - 3 x + 1)$ $x m u s t b e a r o o t x^{2} - 3 x + 1 = 0$ $\Rightarrow x = \frac{3 + \sqrt{5}}{2} > 2 o r x = \frac{3 - \sqrt{5}}{2} < 2$ $x = \sqrt[8]{2207 - \frac{1}{2207 - x^{8}}} > \sqrt[8]{2207 - \frac{1}{2207}} > \sqrt[8]{256} = 2$ $s o t h e s o l u t i o n i s x = \frac{3 + \sqrt{5}}{2}$
	Answered by MJS_new last updated on 06/Jan/21

	$x = 2207 - \frac{1}{x} \Rightarrow x = \frac{2207 + 987 \sqrt{5}}{2}$ $\sqrt{x} = \frac{47 + 21 \sqrt{5}}{2}$ $\sqrt[4]{x} = \sqrt{\sqrt{x}} = \frac{7 + 3 \sqrt{5}}{2}$ $\sqrt[8]{x} = \sqrt{\sqrt{\sqrt{x}}} = \frac{3 + \sqrt{5}}{2}$ $each step like this : {(a + b \sqrt{5})}^{2} = \frac{α + β \sqrt{5}}{2}$ $by matching the constants$
	Commented by mr W last updated on 06/Jan/21

	$c a n w e a n d s h o u l d w e g e t t w o p o s s i b l e$ $s o l u t i o n s \frac{3 \pm \sqrt{5}}{2} ?$
	Commented by MJS_new last updated on 06/Jan/21

	$2207 - \frac{1}{2207} \approx 2207$ $x = 2207 - \frac{1}{x} \Rightarrow x \approx 2207 \lor x \approx \frac{1}{2207}$ $\Rightarrow only one solution$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum